Automorphisms of Generalized Fermat manifolds

Abstract

Let $d\ge 1$, $k\ge 2$ and $n\ge d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $\ddot{I}€:M\to {\mathbb{P}}^d$, with deck group $H\cong {\mathbb{Z}}_k^n$, whose branch divisor consists of $n+1$ hyperplanes in general position, each one of branch order $k$. In this case, $H$ is called a generalized Fermat group of type $(d;k,n)$. In previous work, we proved that the generalized Fermat group $H$ is unique in the following cases: (i) $d=1$ and $(k-1)(n-1)>2$, or (ii) $d\ge 2$ and $(d;k,n)\notin \{(2;2,5),(2;4,3)\}$. To obtain this uniqueness fact, we used a differential method due to Kontogeorgis. This paper provides a different and shorter proof of the uniqueness of $H$. We also study the locus of fixed points of subgroups of $H$.

corrections from previous version and shorter version